*Classical Tessellations and 3-Manifolds* Spring 2014

*Topics in Geometry; Math 24100*

### Instructor: Danny Calegari

### MWF 12:30-1:20; Pick 22

### Description of course:

This course explores the symmetries of classical tessellations --- e.g. wallpaper patterns,
friezes, crystals, especially in 2 dimensions --- and their connections to group theory and
topology, especially the topology of 3-dimensional manifolds.

(Note: because of a lack of time, we never got to
3-manifolds, so the course concentrated on the connections between groups,
geometry and topology in 2 dimensions)

### Cancellations:

None yet.

### Notices:

Mathematics Department policy says that I should opt out of the online
evaluation process, relying instead on paper evaluations. I plan to comply with
this policy; when paper evaluation forms are available (some time in the 9th
week) I will distribute them to you in class.

### Homework/Midterm/Final:

There will be a midterm and a final. There will also be weekly homework. Homework is posted to this website each
Friday and is due at the *start* of class the following Friday. Late homework will not be accepted.

- Homework 1, due Friday, April 11; also lozenge.eps and dots_array.eps
- Homework 2, due Friday, April 18; also cubic_lattices.eps
- Homework 3, due Friday, April 25; also triple_point.eps and doubletorus.eps (this file was produced with wireframe)
- Homework 4, due Friday, May 2; also spiral_tiling.eps and paper_dolls.eps
- Midterm was given out in class Friday, May 2; also sketch solutions by Yan Mary He
- Homework 5, due Friday, May 16; also V_minus_one.eps
- Homework 6, due Friday, May 23; also Farey.eps
- Homework 7, due Friday, May 30; also right_angled_pentagons.eps
- Final was given out in class Friday, May 30.

### Syllabus:

The skeleton of the syllabus is the following.

- Introduction to Tessellations
- Continuous and discrete groups
- Euclidean Wallpaper groups
- Spherical and hyperbolic geometry
- Surfaces and circle bundles
- Seifert fibered spaces

### Notes from class and downloads:

### References:

The textbook for the class is:

Bill Casselman's online copy of

*Mathematical Illustrations*, an introduction to programming in postscript, is

here.